Volker Pluschke, Frank Weber
The local solution of a parabolic-elliptic equation with a nonlinear Neumann boundary condition

Comment.Math.Univ.Carolinae 40,1 (1999) 13-38.

Abstract:We investigate a parabolic-elliptic problem, where the time derivative is multiplied by a coefficient which may vanish on time-dependent spatial subdomains. The linear equation is supplemented by a nonlinear Neumann boundary condition $-\partial u/\partial \nu _A = g(\cdot ,\cdot ,u)$ with a locally defined, $L_r$-bounded function $g(t,\cdot ,\xi )$. We prove the existence of a local weak solution to the problem by means of the Rothe method. A uniform a priori estimate for the Rothe approximations in $L_{\infty }$, which is required by the {local} assumptions on $g$, is derived by a technique due to J. Moser.

Keywords: parabolic-elliptic problem, nonlinear Neumann boundary condition, Rothe method
AMS Subject Classification: 35K65, 65N40, 35M10